slides created by Marty Stepp and Hélène Martin CSE 143 Lecture 23 Hashing read 11.2 slides created by Marty Stepp and Hélène Martin http://www.cs.washington.edu/143/

SearchTree as a set We implemented a class SearchTree to store a BST of ints: Our BST is essentially a set of integers. Operations we support: add contains remove ... But there are other ways to implement a set... 91 60 87 29 55 42 -3 overallRoot

How to implement a set? Elements of a TreeSet (IntTree) are in BST sorted order. We need this in order to add or search in O(log N ) time. But it doesn't really matter what order the elements appear in a set, so long as they can be added and searched quickly. Consider the task of storing a set in an array. What would make a good ordering for the elements? index 1 2 3 4 5 6 7 8 9 value 11 24 49 index 1 2 3 4 5 6 7 8 9 value 11 24 49

Hashing hash: To map a value to an integer index. hash table: An array that stores elements via hashing. hash function: An algorithm that maps values to indexes. one possible hash function for integers: HF(I)  I % length set.add(11); // 11 % 10 == 1 set.add(49); // 49 % 10 == 9 set.add(24); // 24 % 10 == 4 set.add(7); // 7 % 10 == 7 index 1 2 3 4 5 6 7 8 9 value 11 24 49

Efficiency of hashing Add: set elementData[HF(i)] = i; public static int HF(int i) { return Math.abs(i) % elementData.length; } Add: set elementData[HF(i)] = i; Search: check if elementData[HF(i)] == i Remove: set elementData[HF(i)] = 0; What is the runtime of add, contains, and remove? O(1)! Are there any problems with this approach?

Collisions collision: When hash function maps 2 values to same index. set.add(11); set.add(49); set.add(24); set.add(7); set.add(54); // collides with 24! collision resolution: An algorithm for fixing collisions. index 1 2 3 4 5 6 7 8 9 value 11 54 49

Probing probing: Resolving a collision by moving to another index. linear probing: Moves to the next index. set.add(11); set.add(49); set.add(24); set.add(7); set.add(54); // collides with 24; must probe Is this a good approach? variation: quadratic probing moves increasingly far away index 1 2 3 4 5 6 7 8 9 value 11 24 54 49

Clustering clustering: Clumps of elements at neighboring indexes. slows down the hash table lookup; you must loop through them. set.add(11); set.add(49); set.add(24); set.add(7); set.add(54); // collides with 24 set.add(14); // collides with 24, then 54 set.add(86); // collides with 14, then 7 Now a lookup for 94 must look at 7 out of 10 total indexes. index 1 2 3 4 5 6 7 8 9 value 11 24 54 14 86 49 index 1 2 3 4 5 6 7 8 9 value

Chaining chaining: Resolving collisions by storing a list at each index. add/search/remove must traverse lists, but the lists are short impossible to "run out" of indexes, unlike with probing index 1 2 3 4 5 6 7 8 9 value 11 24 7 49 54 14

Hash set code import java.util.*; // for List, LinkedList public class HashIntSet { private static final int CAPACITY = 137; private List<Integer>[] elements; // constructs new empty set public HashSet() { elements = (List<Integer>[]) (new List[CAPACITY]); } // adds the given value to this hash set public void add(int value) { int index = hashFunction(value); if (elements[index] == null) { elements[index] = new LinkedList<Integer>(); elements[index].add(value); // hashing function to convert objects to indexes private int hashFunction(int value) { return Math.abs(value) % elements.length; ...

Hash set code 2 ... // Returns true if this set contains the given value. public boolean contains(int value) { int index = hashFunction(value); return elements[index] != null && elements[index].contains(value); } // Removes the given value from the set, if it exists. public void remove(int value) { if (elements[index] != null) { elements[index].remove(value);

Rehashing rehash: Growing to a larger array when the table is too full. Cannot simply copy the old array to a new one. (Why not?) load factor: ratio of (# of elements ) / (hash table length ) many collections rehash when load factor ≅ .75 can use big prime numbers as hash table sizes to reduce collisions 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 24 7 49 11 54 14

Rehashing code ... // Grows hash array to twice its original size. private void rehash() { List<Integer>[] oldElements = elements; elements = (List<Integer>[]) new List[2 * elements.length]; for (List<Integer> list : oldElements) { if (list != null) { for (int element : list) { add(element); }

Other questions How would we implement toString on a HashSet? How would we implement an Iterator over a HashSet? index 1 2 3 4 5 6 7 8 9 value 11 24 7 49 current element: 24 current index: 4 sub-index: 0 iterator 54 14

Hashing objects It is easy to hash an integer I (use index I % length ). How can we hash other types of values (such as objects)? All Java objects contain the following method: public int hashCode() Returns an integer hash code for this object. We can call hashCode on any object to find its preferred index. How is hashCode implemented? Depends on the type of object and its state. Example: a String's hashCode adds the ASCII values of its letters. You can write your own hashCode methods in classes you write. All classes come with a default version based on memory address.

Hash function for objects public static int HF(E e) { return Math.abs(e.hashCode()) % elements.length; } Add: set elements[HF(o)] = o; Search: check if elements[HF(o)].equals(o) Remove: set elements[HF(o)] = null;

String's hashCode The hashCode function inside String objects looks like this: public int hashCode() { int hash = 0; for (int i = 0; i < this.length(); i++) { hash = 31 * hash + this.charAt(i); } return hash; As with any general hashing function, collisions are possible. Example: "Ea" and "FB" have the same hash value. Early versions of the Java examined only the first 16 characters. For some common data this led to poor hash table performance.

Implementing a hash map A hash map is just a set where the lists store key/value pairs: // key value map.put("Marty", 14); map.put("Jeff", 21); map.put("Kasey", 20); map.put("Stef", 35); Instead of a List<Integer>, write an inner Entry node class with key and value fields; the map stores a List<Entry> index 1 2 3 4 5 6 7 8 9 value "Stef" 35 "Marty" 14 "Jeff" 21 "Kasey" 20

Implementing a tree map Similar to difference between HashMap and HashSet: Each node now will store both a key and a value tree is BST ordered by keys keys must be Comparable overall root key = "Kate" val = 28 key = "Jack" val = 36 key = "Sayid" val = 38 key = "Desmond" val = 49 key = "Locke" val = 51 key = "Sawyer" val = 34